Abstract:We derive explicit positive $e_I$-expansions for the chromatic symmetric functions of several graph families. First, we obtain a uniform formula for KPKP graphs that specializes to known formulas for lollipops, KPK graphs, KKP graphs, and PKP graphs. We then give positive $e_I$-expansions for twinned paths and cycles and apply the KPKP formula to twinned lollipops. Finally, we establish a positive $e_I$-expansion for kayak paddles and specialize it to infinity graphs. These formulas refine ordinary $e$-positivity by retaining composition-indexed coefficients and provide independent proofs complementary to existing recurrence, unit-interval, and noncommutative methods.
From: Davion Q. B. Tang [view email]
[v1]
Fri, 2 Aug 2024 16:42:08 UTC (56 KB)
[v2]
Sat, 16 Nov 2024 04:12:02 UTC (24 KB)
[v3]
Thu, 6 Aug 2026 08:09:08 UTC (73 KB)
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